Properties Of ∩-structures

∩-structures and the closure systems (i.e. topped∩-strcutures) are related to many realms, such as mathematics and computer science. In this paper, definitions of basis, remote neighborhood basis, sum, product and contin-uous mapping, dense subset, weight, character and density on∩-structures are presented, and their corresponding properties are discussed in addition. Further-more, it is proved that a closure systerm on a given set X can be determined by any operator of the following ones:a weak closure operator, a weak interior operator, a weak exterior operator, a weak boundary operator, a weak derived operator, a weak difference derived operator, a weak remote neighborhood system operator, or a weak neighborhood system operator.The key points and the main contents of this paper are as follows:In Chapter one, some concepts of convergence and important conclusions which are related to∩-structure and closure system are introduced.In Chapter two, according to the topology method, definitions of basis, sum and product are given first. Then an intensive approach of the weight of∩-structures is studied, and some results similar to the topological space are obtained consequently. Finally, it is proved that every finite∩-structure has the smallest base, and an algorithm to obtain the smallest base and related program and example are also given.In Chapter three, definitions of remote neighborhood basis, continuous map-ping, dense subset, character and density on∩-structures are given first, and then the relation between the weight, character and density on∩-structures are dis-cussed. Finally, the character and density on a class of∩-structures and their product∩-structures are dicussed.In Chapter four, some appropriate order relations on WCL(X) (the set of all weak closure operators), WIN(X) (the set of all weak interior operators), WOU(X) (the set of all weak exterior operators), WB(X) (the set of all weak boundary oper-ators), WD(X) (the set of all weak derived operators), WD*(X) (the set of all weak difference derived operators), WR(X) (the set of all weak remote neighborhood sys-tem operators) and WN(X) (the set of all weak neighborhood system operators) are defined, respectively, for an abitrary set X. It is proved that the relations defined above correspondingly make WCL(X), WIN(X), WOU(X), WB(X), WD(X), WD*(X), WR(X) and WN(X) become complete lattices, which are ismorphic to (CS(X), (?)), where CS(X) is the set of all closure systerms on X.